Precalculus Examples

$2 + 13 \sin (x) = 14 \cos^{2} (x)$

Step 1

Subtract $14 \cos^{2} (x)$ from both sides of the equation.

$2 + 13 \sin (x) - 14 \cos^{2} (x) = 0$

Step 2

Replace the $- 14 \cos^{2} (x)$ with $- 14 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$2 + 13 \sin (x) - 14 (1 - \sin^{2} (x)) = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$2 + 13 \sin (x) - 14 \cdot 1 - 14 (- \sin^{2} (x)) = 0$

Step 3.2

Multiply $- 14$ by $1$ .

$2 + 13 \sin (x) - 14 - 14 (- \sin^{2} (x)) = 0$

Step 3.3

Multiply $- 1$ by $- 14$ .

$2 + 13 \sin (x) - 14 + 14 \sin^{2} (x) = 0$

Step 4

Subtract $14$ from $2$ .

$13 \sin (x) - 12 + 14 \sin^{2} (x) = 0$

Step 5

Reorder the polynomial.

$14 \sin^{2} (x) + 13 \sin (x) - 12 = 0$

Step 6

Substitute $u$ for $\sin (x)$ .

$14 {(u)}^{2} + 13 (u) - 12 = 0$

Step 7

Factor by grouping.

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Step 7.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 14 \cdot - 12 = - 168$ and whose sum is $b = 13$ .

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Step 7.1.1

Factor $13$ out of $13 u$ .

$14 u^{2} + 13 u - 12 = 0$

Step 7.1.2

Rewrite $13$ as $- 8$ plus $21$

$14 u^{2} + (- 8 + 21) u - 12 = 0$

Step 7.1.3

Apply the distributive property.

$14 u^{2} - 8 u + 21 u - 12 = 0$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$14 u^{2} - 8 u + 21 u - 12 = 0$

Step 7.2.2

Factor out the greatest common factor (GCF) from each group.

$2 u (7 u - 4) + 3 (7 u - 4) = 0$

Step 7.3

Factor the polynomial by factoring out the greatest common factor, $7 u - 4$ .

$(7 u - 4) (2 u + 3) = 0$

Step 8

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$7 u - 4 = 0$

$2 u + 3 = 0$

Step 9

Set $7 u - 4$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $7 u - 4$ equal to $0$ .

$7 u - 4 = 0$

Step 9.2

Solve $7 u - 4 = 0$ for $u$ .

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Step 9.2.1

Add $4$ to both sides of the equation.

$7 u = 4$

Step 9.2.2

Divide each term in $7 u = 4$ by $7$ and simplify.

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Step 9.2.2.1

Divide each term in $7 u = 4$ by $7$ .

$\frac{7 u}{7} = \frac{4}{7}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $7$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{7 u}{7} = \frac{4}{7}$

Step 9.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{4}{7}$

Step 10

Set $2 u + 3$ equal to $0$ and solve for $u$ .

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Step 10.1

Set $2 u + 3$ equal to $0$ .

$2 u + 3 = 0$

Step 10.2

Solve $2 u + 3 = 0$ for $u$ .

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Step 10.2.1

Subtract $3$ from both sides of the equation.

$2 u = - 3$

Step 10.2.2

Divide each term in $2 u = - 3$ by $2$ and simplify.

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Step 10.2.2.1

Divide each term in $2 u = - 3$ by $2$ .

$\frac{2 u}{2} = \frac{- 3}{2}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{- 3}{2}$

Step 10.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 3}{2}$

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{3}{2}$

Step 11

The final solution is all the values that make $(7 u - 4) (2 u + 3) = 0$ true.

$u = \frac{4}{7}, - \frac{3}{2}$

Step 12

Substitute $\sin (x)$ for $u$ .

$\sin (x) = \frac{4}{7}, - \frac{3}{2}$

Step 13

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{4}{7}$

$\sin (x) = - \frac{3}{2}$

Step 14

Solve for $x$ in $\sin (x) = \frac{4}{7}$ .

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Step 14.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (\frac{4}{7})$

Step 14.2

Simplify the right side.

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Step 14.2.1

Evaluate $\arcsin (\frac{4}{7})$ .

$x = 0.60824557$

Step 14.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = (3.14159265) - 0.60824557$

Step 14.4

Solve for $x$ .

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Step 14.4.1

Remove parentheses.

$x = 3.14159265 - 0.60824557$

Step 14.4.2

Remove parentheses.

$x = (3.14159265) - 0.60824557$

Step 14.4.3

Subtract $0.60824557$ from $3.14159265$ .

$x = 2.53334707$

Step 14.5

Find the period of $\sin (x)$ .

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Step 14.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 14.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 14.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 14.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.60824557 + 2 π n, 2.53334707 + 2 π n$ , for any integer $n$

Step 15

Solve for $x$ in $\sin (x) = - \frac{3}{2}$ .

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Step 15.1

The range of sine is $- 1 \leq y \leq 1$ . Since $- \frac{3}{2}$ does not fall in this range, there is no solution.

No solution

Step 16

List all of the solutions.

$x = 0.60824557 + 2 π n, 2.53334707 + 2 π n$ , for any integer $n$